In analytic geometry, any
equation involving the coordinates specifies a
subset of the plane, namely the
solution set for the equation, or
locus. For example, the equation
y =
x corresponds to the set of all the points on the plane whose
x-coordinate and
y-coordinate are equal. These points form a
line, and
y =
x is said to be the equation for this line. In general, linear equations involving
x and
y specify lines,
quadratic equations specify
conic sections, and more complicated equations describe more complicated figures. Usually, a single equation corresponds to a
curve on the plane. This is not always the case: the trivial equation
x =
x specifies the entire plane, and the equation
x2 +
y2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a
surface, and a curve must be specified as the
intersection of two surfaces (see below), or as a system of
parametric equations. The equation
x2 +
y2 =
r2 is the equation for any circle centered at the origin (0, 0) with a radius of r.
Lines and planes Lines in a
Cartesian plane, or more generally, in
affine coordinates, can be described algebraically by
linear equations. In two dimensions, the equation for non-vertical lines is often given in the
slope-intercept form: y = mx + b where: •
m is the
slope or
gradient of the line. •
b is the
y-intercept of the line. •
x is the
independent variable of the function
y =
f(
x). In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the
normal vector) to indicate its "inclination". Specifically, let \mathbf{r}_0 be the position vector of some point P_0 = (x_0,y_0,z_0), and let \mathbf{n} = (a,b,c) be a nonzero vector. The plane determined by this point and vector consists of those points P, with position vector \mathbf{r}, such that the vector drawn from P_0 to P is perpendicular to \mathbf{n}. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points \mathbf{r} such that \mathbf{n} \cdot (\mathbf{r}-\mathbf{r}_0) =0. (The dot here means a
dot product, not scalar multiplication.) Expanded this becomes a (x-x_0)+ b(y-y_0)+ c(z-z_0)=0, This is just a
linear equation: ax + by + cz + d = 0, \text{ where } d = -(ax_0 + by_0 + cz_0). Conversely, it is easily shown that if
a,
b,
c and
d are constants and
a,
b, and
c are not all zero, then the graph of the equation ax + by + cz + d = 0, {{cn span|text=is a plane having the vector \mathbf{n} = (a,b,c) as a normal.|date=April 2022}} This familiar equation for a plane is called the
general form of the equation of the plane. In three dimensions, lines can
not be described by a single linear equation, so they are frequently described by
parametric equations: x = x_0 + at y = y_0 + bt z = z_0 + ct where: •
x,
y, and
z are all functions of the independent variable
t which ranges over the real numbers. • (
x0,
y0,
z0) is any point on the line. •
a,
b, and
c are related to the slope of the line, such that the
vector (
a,
b,
c) is parallel to the line.
Conic sections In the
Cartesian coordinate system, the
graph of a
quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0\text{ with }A, B, C\text{ not all zero.} As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional
projective space \mathbf{P}^5. The conic sections described by this equation can be classified using the
discriminant B^2 - 4AC . If the conic is non-degenerate, then: • if B^2 - 4AC , the equation represents an
ellipse; • if A = C and B = 0 , the equation represents a
circle, which is a special case of an ellipse; • if B^2 - 4AC = 0 , the equation represents a
parabola; • if B^2 - 4AC > 0 , the equation represents a
hyperbola; • if we also have A + C = 0 , the equation represents a
rectangular hyperbola.
Quadric surfaces A
quadric, or
quadric surface, is a
2-dimensional
surface in 3-dimensional space defined as the
locus of
zeros of a
quadratic polynomial. In coordinates , the general quadric is defined by the
algebraic equation \sum_{i,j=1}^{3} x_i Q_{ij} x_j + \sum_{i=1}^{3} P_i x_i + R = 0. Quadric surfaces include
ellipsoids (including the
sphere),
paraboloids,
hyperboloids,
cylinders,
cones, and
planes. ==Distance and angle==